To simplify the given fraction [tex]\(\frac{2x^2 + 9x - 5}{x^2 + 2x - 15}\)[/tex], we need to factor both the numerator and the denominator and then reduce the fraction by canceling out any common factors. Here is the step-by-step process:
1. Factor the denominator [tex]\(x^2 + 2x - 15\)[/tex]:
- We need to find two numbers that multiply to [tex]\(-15\)[/tex] (the constant term) and add up to [tex]\(2\)[/tex] (the coefficient of the linear term [tex]\(x\)[/tex]).
- These numbers are [tex]\(5\)[/tex] and [tex]\(-3\)[/tex], because [tex]\(5 \cdot (-3) = -15\)[/tex] and [tex]\(5 + (-3) = 2\)[/tex].
- Thus, we can factor the denominator as:
[tex]\[
x^2 + 2x - 15 = (x + 5)(x - 3)
\][/tex]
2. Factor the numerator [tex]\(2x^2 + 9x - 5\)[/tex]:
- We need to find two numbers that multiply to [tex]\(2 \cdot (-5) = -10\)[/tex] and add up to [tex]\(9\)[/tex].
- These numbers are [tex]\(10\)[/tex] and [tex]\(-1\)[/tex], because [tex]\(10 \cdot (-1) = -10\)[/tex] and [tex]\(10 + (-1) = 9\)[/tex].
- We can then break up the middle term [tex]\(9x\)[/tex] into [tex]\(10x - x\)[/tex] to help in factoring by grouping:
[tex]\[
2x^2 + 9x - 5 = 2x^2 + 10x - x - 5
\][/tex]
- Group the terms as follows:
[tex]\[
2x(x + 5) - 1(x + 5)
\][/tex]
- Factor out the common term [tex]\((x + 5)\)[/tex]:
[tex]\[
(2x - 1)(x + 5)
\][/tex]
3. Write the fraction with the factored numerator and denominator:
[tex]\[
\frac{2x^2 + 9x - 5}{x^2 + 2x - 15} = \frac{(2x - 1)(x + 5)}{(x + 5)(x - 3)}
\][/tex]
4. Cancel the common factors [tex]\(x + 5\)[/tex] in the numerator and denominator:
[tex]\[
\frac{(2x - 1)(x + 5)}{(x + 5)(x - 3)} = \frac{2x - 1}{x - 3}
\][/tex]
Therefore, the simplified form of the fraction [tex]\(\frac{2x^2 + 9x - 5}{x^2 + 2x - 15}\)[/tex] is:
[tex]\[
\frac{2x - 1}{x - 3}
\][/tex]
